Free Energy Subadditivity for Symmetric Random Hamiltonians

Abstract

We consider a random Hamiltonian $H:\Sigma \to \mathbb{R}$ defined on a compact space $\Sigma$ that admits a transitive action by a compact group $\mathcal{G}$. When the law of $H$ is $\mathcal{G}$-invariant, we show its expected free energy relative to the unique $\mathcal{G}$-invariant probability measure on $\Sigma$ obeys a subadditivity property in the law of $H$ itself. The bound is often tight for weak disorder and relates free energies at different temperatures when $H$ is a Gaussian process. Many examples are discussed including branching random walk, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum SK and SYK models.

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